L Crane, FW Lawvere, P Cartier, C McLarty (contd.)

0:00 There are two halves to the function of mathematics. The first problem is that I know it goes farther than I have been able to do. The second half is for me to be able to process and understand those things. But, he said, well, would you take a chance with the first part only if you had to slow down the second part and keep on with the first part to accept it? So I take it. I'm like, hey, I said Omnevei. No, no, no, no, no. Omnevei was, what was his name, a journal. Like Carlos, of Carlos. Yeah, yeah. Of course he had good feelings, and he would take advice from him, but he was not a journalist. Yes, yes, yes. Of course. It just takes an extremely intelligent autocrat. Very intelligent and also, of course, very disciplined and energetic autocrat. And I remember when I had the fact about the script one day, I promised to myself that... In a not-too-distant future, I would be able to read the second half of the world in the last. Understand that? Good. I did. That was one of my motivations to study programming. The second one was that when I worked out that in programming, they offered me a teaching job for my time. I thought that would be the most good of life. I look at them, I say, well, I do, I look at them, and I try to divide them, you know, all the people, to purely abstract the people of the time and the old-fashioned television. Yes, something which would have applications from, you know, the value for those. But my main focus is to teach a new concept, very powerful, and not too abstract. Because if you read the tubes, you'll know. This, of course, is the machinery that they are. This is just a stochastic.

2:30 Analysis is the machinery that they apply in financial economics, so it's all about deciding when to place a put or call option, the derivative, and things like that. Just to formalize the idea that the information is growing in time. So it's growing in time, yes. So there's going to be measures on it. Yes. And then I'm not sure. The probability estimates actually may depend on the previous information. But very, very, very limited. But I made a great effort to translate it into pragmatic terms. So that's four systems of physics. And what I find in the best book is the fellow. I think I'm the fellow. At that time there were three majors, but I felt safe. Go ahead. Though both are quite abstract, and Federer is absolutely excellent in the first volume, where it is his normal self and his way of looking at probability, and he has a deep sense of what is change. The second volume is more analytical and formal, and the first volume is fantastic. The second volume is, in a sense, he wants to come back, to come to Trojans as ten more standard approaches. That's not good. Yes, true. What, the more standard approach being the Commodoro? And so, and then, and then when I joined Sparta, of course, there was a time when the University of New York,

5:00 Start with products. And with the discussion of microbeads and so on. So I said, OK, geometry, algebra, topology is part of the traditional. You should open the way you have used. And then I said, why not the whole thing? Because for me, that must be that quality. It was brilliant that Rosemary neglected Rosemary. Yeah. Well, he had been honest since Pharrell. Oh, yeah. Well, I have no idea. Well, mind you, what Varelda did was so tremendous in terms of systematizing, synthesizing. At that time, I guess, the people were, yeah, that's exactly, it's 40 years behind. Oh, I'm good. But you're right, very little bit of time from the new class on since then. And yet then there was an explosion of work at a very high level, including much of the school of geometric modernization, which of course was done by French mathematicians, or people working in France, if not Constantin. That, uh, that, uh, probably made for some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh, some of the, uh. Armament des toboggans, vérification de la porte opposée. They took the modificable momentum out of the 3-2-3. That was the great caesura in society and politics.

7:30 You had to taste that. And I remember that part, when I was told by you. Madame, Monsieur, nous vous rappelons que ce vol est non fumeur et nous informons que l'utilisation d'appareils électroniques est interdite durant le décollage et l'atterrissage. Les démonstrations de sécurité vont vous être présentées. Accordez-nous quelques instants d'attention. Ladies and gentlemen, we remind you that this is an unswacking flight. We inform you that the use of electronic devices is not allowed during takeoff and landing. We will now show you the safety procedures. May we please have your attention for a few moments. It's interesting. I was taught this story by my grandfather. And my grandfather has died just in the military. He was a military, but not by choice. He was also one of the various director of the University of Chicago, so that was very common at the time, but he was not an enthusiast in the future, and in the beginning of the century, he was a guy who was on the platform, where he might have caused a lot of trouble. The speakers want us to watch the security thing. Yes, and of course they are in an army where the officer corps was almost solidly under the sun. And so there were two occasions when there was a worker strike and the army. And my grandfather did all of them. It was his power to prevent crashes. And he came back home by singing the Internationale. Good for him. But that wasn't something which many French army officers did. No. And then, during this, he was provoked by one of his fellow wives to do a duel. Oh, to do a duel.

10:00 It was just a play. It was not a space battle. Yeah, it took time. He was challenged by his friend when he did it. Both were arrested for thinking on the ice for a month or something like that. In fact, he was trapped. And, like, but he died. He was killed. Of course, the casualties of the first days were the highest of the whole war in terms of war. I was reading this extremely fine essay about Gatteau. ...who was a student of... well, he did some work with Hadamard, but he had just gone to Rome, and he studied with Voltaire, but then came back in 1913. No, that's right, he went in 1913 to study with Voltaire, and came back to Paris the following year, and was clearly going to be one of the most brilliant mathematicians of his generation. He was killed on the 22nd of August, which was in fact the day that there were some more casualties. On any other single day of the entire war, more than any army, more than the British, That was total casualties, not just the kill. That was more of a British loss on the Somme on the first day, and the British tell themselves confidently that was the greatest, that their losses on that first day of the Somme, which were 60,000 in a single day, in a single morning, were the greatest casualties that any army has ever suffered in a single day of fighting. In fact, the French army exceeded those losses in one day in August 1920. It was the Battle of the Frontiers. It was just this insane... ...strategy of just marching in parade ground formations, straight into the German machine gun, which is just unbelievable. And in fact, you only have to look at any of the war memorials in any French town or village, and every French town or village has a war memorial. Where the names are actually shown by the year, which they frequently are, the casualties for 1940 are almost invariably longer than those of any of the other years that you've heard.

12:30 There is only four months of fighting in there. That's against Fulham and all the other years in 1918. But the casualties in 1940 are unusually confident. The barge at the post for any other year, the barge at the post for the year of Bogdanov in 1915, it's astonishing. The casualties at the very outset of the war were just unbelievable. Reading Gatteau's letters to Volterra... She wrote quite a few novels on battlefields. It's quite an extraordinary movie. I must send you the article. It's called The Ghosts of the Equinox. It also has a little bit of the second part of the article about André Leung. I'm trying to remember the name of the author. I think you can locate it. Yes, no, I think it's quite interesting. It's an English writer, but one who has studied narrative. I will dig it out when I get back home. I think I'm going to try and get everyone to sleep if I may. I want you to make your partner quite busy. No, just the same. It's perfect timing. As soon as we're able I can put the seat back. Well, in fact, I am. We should be able to get some rest, depending on the turbulence.

20:00 Thank you for your attention.

22:30 I don't think that I haven't talked to him in private, because that depends on the people, you know. Colin, for instance, he will be here after. And we have asked several people in Europe anyway around that day. So it looks like it's definitely going to end. What? The question of what the whole thing looks like. No, no, it depends on the overall decision, you know. Well, it could be elsewhere, but then we would need more funding. Because then we would need funds, not only in region two. Well, we've got some support positions in the discipline of mathematics, from the, I know it sounds as if, you know, you've got quite a couple of colleagues in the centre for Einstein. Well, they might be the answer to that question. Um, that's the question, yeah. I think that I wasn't invited. Yeah, but we didn't have enough money. I'm really sorry. In the end, we ended up with just 3,000. I'm sorry, but we ended up with a budget of only $5,000, I think, depending on the time you pay for your seat at the hotel or something like that. If you do want to get started in all this, then you can put me in the box and then you can get started with anything you want. So I'm hoping that next year we might have some money, and I'm actually going to try that tomorrow when I go to Belkin to speak to the Royal Academy there, that we might have some additional funding for the Royal Fund, because we've been very happy to be meeting you, and it's important to be meeting you.

25:00 Yes, well, it's the same story, having to wait a couple of weeks, I have to put my multiplication in front of my proposal. On the paper, in order to send it to the people first, telling them this is the possible date... Are you interested? Are you perhaps already in Europe around that stage? Do you have a list of the people you intend to invite? Yes, sure, sure. Well, can you give it to me because I'm going to be... But by May tomorrow. Yeah, but it's not... No, no, I don't mean right now, but it's... Because I'm going to tell you that I'm going to be talking to an economist tomorrow. Well, I'm actually going to be talking to an entrepreneur on Thursday that I'm doing. And this is a general application. Well, the thing is... Hang on, I'm making a general application. Funding for the archive may include projects for digitizing classes of the online. But they want to investigate, they are interested in having these types of things, to have activities for the next year. And if I can show you, well this is not something that we would like to give support for, because it may make a difference, whether it's happening or not happening, it would be helpful to have a list of, you know, intents and purposes. That would be great. That would be great, that would be great. Thank you for your attention.

27:30 Because it was originally intended to be, one part, the subject was the representation of space, and the first part was supposed to be pure mathematics, and basically the legacy of Rosenzweig and algebraic geometry, and obviously beyond, there was an algebraic geometry of the rest of mathematics, and of course, the description of spectra, and of course, of mathematics, and obviously, of course, of mathematics. And the second part was supposed to be... ...application of the category of theory and physics. But unfortunately, because of the pandemic, we only had one major talk on the subject of physics, which was the crane. So, you know, we had the crane, and there was a list of all the things we were going to be talking about. And we became pretty cool with that. The report was hung up on Twitter by the Nobel-pennsylb category, and I will grab the floor again. So, Matt and Bill attack Kevin about what they think about mathematics and arithmetic. We will let them talk about mathematics later on. So do that, because Bill is going to start a debate with Carter. On what point did he attack him? This little here helps in a very strong view that there is a category in which you can have, you know, this geometry has got to be kind of like a combination of those categories. The closed-ended category is the first. It's given to the third layer. The non-mathematic category is the first category. And without this, if you've got that, you'll lose the reader equipment. This is the reason. But he made this the reason for this. Then there's the usual kind of refining of the non-convective geometry argument that non-convective geometry will be equal, well, it's better than non-convective, no, this is not what we're talking about. The usual refining of the non-convective geometry is the form that you can base it on.

30:00 It's extremely impressive. It's a very, very, very good presentation of these three interventions, the first two, the second two, and also one of the great debates around the world. It's an incredible thing. It's very, very good. I came in with him, so therefore, I need some... Yes, that's it. We are going to... But he did what? He did what? He did what? He did what? Thank you for your attention. Also, there are some problems in families with these little kids. For example, the first time, when they were 17 years old, they couldn't answer the phone. No, I understand. But he is an ordinary person. Yes, he is an ordinary person. When he was a child, he needed to answer the phone. I think everything is going well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well, very well,

32:30 Maybe there is still a conclusion in relation to mathematics to know that it is not necessary that this lecture is put in place. We will consider it here in the material table. I will still, between those of you who know the questions well and the others, I will still not answer for an hour on things that... Some of you probably have never heard of them, so I'm going to give you a quick introduction. Then I'm going to talk about the work of Daniel Kahl, who is a mathematician of Dutch origin. In 1956, in New York, Samuel Albert, who came from... Small correction, he is of Dutch origin. He is originally from the Netherlands, not Israel. He came from Israel, yes, of course. Because we spoke yesterday. Okay, thank you. It's a small thing.

35:00 I'm going to talk in more detail about the role of concentration in mathematics in the geosciences of the world. I will also come to the 4th chapter, where I will talk a little about the theory of category theory in general, but also from the second half of the 40s to the end of the 60s. I will talk about the terminology of the world, and I will also talk about the history of the world. And then I will simply talk about a category. I give a definition. You have to understand it first as an axiomatic definition. So we say nothing about the nature of objects and arrows, but we simply have two collections. Even if we talk about the nature of these collections, whether they are children, children of a place or not, it does not matter at the moment. So two collections of objects and arrows. A domain object and a codomain object are written from F to A to B. If you have two arrows like this, the composition is that the codomain F has the property.

37:30 The object has only one element, it is a monoid. There are only arrows. So you can think of a category as a monoproduct to generalize to several objects. But the more classic, more intuitive examples are the three categories there and others of the same range. Namely, you have in the category of sets, you have objects, they are the others together, and the tests are the functions, they are not together, the functions there are directed. In any case, the rule is that topological spaces, continuous functions, groups, normomorphisms, and so on, can be taken in any kind of structure, and there I have already given a term that I will not mention. If we want to make a category, we just have to have a science that respects the structure. An object composed of two functions. The first function is the factor between categories. The object of the category associates an object of the other category, of course, and to any arrow it associates another arrow of the other category. And this in a compatible way with the components either in the original category, or in the category of the other category.

40:00 There, I did not write down the examples for now, but... The key historical example, which was introduced in 1945 by Eilenberg and MacLean, was the concept of homology. You take the category of topological spaces, you look for groups that are invariant for a topological space. If you have a continuous function, a number of them show up in groups. This was the example we used for the first ones, but there are simpler ones that we will use in the future. For example, if you take a group, well, if you take a set, you can make it the basis of a free group. Now, if you don't know the concepts, I'm having a hard time expressing myself. I would like to introduce you to some other concepts that are a little more modern in the field of mathematics, that's clear, like that, axiomatically on two pages, it's not at all motivating. As for the earlier questions, I wrote a book on this question.

42:30 Now, second, this is a way of expressing oneself that is quite important in all this. This is an arrow diagram. We say that this diagram is commutative if we have this equality of components. So if it's equal, if we go through here, through a function, then these two functions, the components, will give the same value as these two here. But if you have another example, it's the same situation. For now, it seems to be a more complicated way than this one to explain it, but as soon as the equation becomes more complicated, the diagram becomes more intuitive. You have a diagram, it has 20 arrows. It's much easier to understand what's going on by looking at the diagram than by looking at the equations. Now, the last notion to introduce before moving on to the notion of adjoining forces is the notion of force-homomorphism. So you have two forces between the same categories, and here we are interested in the homomorphisms between the forces. In other words, we take a pointer and another and we want to know what to do to go from one to the other. I'm going to spend a lot of time explaining it. A collection of arrows, a collection of arrows. For each object A of A, we have an arrow in it. As for any original arrow in the category of origin, this diagram is as follows. We can either first make the pointer.

45:00 Or we do the transformation first. We have the same result. That's the idea of homomorphism. Also, natural transformation. That's the original American terminology. And now, I'm going to... Excuse me? No, no, no. It's H-A-A. I don't want to make any mistakes. What I'm trying to say is H-A-A. No, no, no. It's not good. There are the A's. Here it's an A3. It's an entry of the category. And so for B, if B is an entry of the same category, I put H1, H2. It's a collection of arrows. So, speaking once again of this construction of a free module, I was talking about a good module, we didn't care, so you take a set and you take it as a base of a module. It is possible, this construction there, and this base has a universal property, this is a term that we will talk about a little later.